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## Baseball and Physics

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**1927 Yankees:**Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physicsteam ever assembled Baseball and Physics MVP’s**#521, September 28, 1960**Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams BA: .344 SA: .634 OBP: .483‡ HR: 521 ‡career record**Introduction: Description of Ball-Bat Collision**• forces large (>8000 lbs!) • time is short (<1/1000 sec!) • ball compresses, stops, expands • kinetic energy potential energy • bat compresses ball….ball bends bat • hands don’t matter! • GOAL: maximize ball exit speed vf • vf 105 mph x 400 ft x/vf = 4-5 ft/mph How to predict vf?**Bat Rest Frame**“Lab” Frame vball vbat vrel vf eAvrel Kinematics: Reference Frames vf =eAvball +(1+eA)vbat • eA “Apparent Coefficient of Restitution” = “BESR” - 0.5 • property of ball & bat • weakly dependent on vrel • 0.2 vf 0.2 vball + 1.2 vbat Conclusion:vbatmuch more important than vball**r bat recoil factor = mball/mbat,eff**e “Coefficient of Restitution” Kinematics: Conservation Laws (Accounting for eA) v m1 m2 eAv m1**.**. . CM b Kinematics: bat recoil factor • typical numbers • mball = 5.1 oz • mbat = 31.5 oz • k = 9.0 in • b = 6.3 in • r = .24 • e = 0.5 • eA = 0.21 = + 0.16 x 1.49 0.24 • All things equal, want r small • But….**All things are not equal**• Mass & Mass Distribution affect bat speed • Conclusion: • mass of bat matters….but probably not a lot**“bounciness” of ball**Kinematics: Coefficient of Restitution (e): (Energy Dissipation) • in CM frame: Ef/Ei = e2 • massive rigid surface: e2 = hf/hi • typically e 0.5 • ~3/4 CM energy dissipated! • probably depends on impact speed • depends on ball and bat!**COR: Is the Ball “Juiced”?**• MLB:e = 0.546 0.032 @ 58 mph on massive rigid surface For ball on stationary bat:**CM**12 10 8 6 4 2 0 -2 0 5 10 15 20 25 30 Putting it all together….. vf = eA vball + (1+eA) vbat**CM**More Realistic Analysis vf = eA vball + (1+eA) vbat**III. Dynamics Model for Ball-Bat Colllision:**Accounting for Energy Dissipation • Collision excites bending vibrations in bat • Ouch!! Thud!! • Sometimes broken bat • Energy lost lower vf (lower e) • Bat not rigid on time scale of collision • What are the relevant degrees of freedom? see AMN, Am. J. Phys, 68, 979 (2000)**The Essential Physics: A Toy Model**bat ball Mass= 1 2 4 rigid << 1 m on Ma (1 on 2) >> 1 m on Ma+Mb (1 on 6) flexible**A Dynamic Model of the Bat-Ball Collision**y 20 Euler-Bernoulli Beam Theory‡ y z • Solve eigenvalue problem for free oscillations (F=0) • normal modes(yn, n) • Model ball-bat force F • Expand y in normal modes • Solve coupled equations of motion for ball, bat ‡Note for experts: full Timoshenko (nonuniform) beam theory used**f1 = 177 Hz**f3 = 1179 Hz f2 = 583 Hz f4 = 1821 Hz nodes Normal Modesof the Bat Louisville Slugger R161 (33”, 31 oz) Can easily be measured (modal analysis)**Measurements via Modal Analysis**Louisville Slugger R161 (33”, 31 oz) FFT frequencybarrel node ExptCalcExptCalc 17917726.526.6 58258327.828.2 1181117929.029.2 1830182130.029.9 Conclusion: free vibrations of bat can be well characterized**F=kxn**F=kxm Model for the Ball 3-parameter problem: k nv-dependence of m COR of ball with rigid surface**Putting it all together….**ball compression • Procedure: • specify initial conditions • numerically integrate coupled equations • find vf = ball speed after ball and bat separate**General Result**energy in nth mode Fourier transform Conclusion:only modes with fn < 1 strongly excited**Results: Ball Exit Speed**Louisville Slugger R161 33-inch/31-oz. wood bat only lowest mode excited lowest 4 modes excited Conclusion:essential physics under control**CM**nodes Application to realistic conditions: (90 mph ball; 70 mph bat at 28”)**The “sweet spot”**1. Maximum vf (~28”) 2. Minimum vibrational energy (~28”) 3. Node of fundamental (~27”) 4. Center of Percussion (~27”) 5. “don’t feel a thing”**3**Displacement at 5” 2 1 y (mm) 0 -1 impact at 27" -2 -3 0 0.5 1 1.5 2 t (ms) Boundary conditions • Conclusions: • size, shape, boundary conditions at far end don’t matter • hands don’ t matter!**T= 0-1 ms**Time evolution of the bat T= 1-10 ms**Wood versus Aluminum**• Kinematics • Length, weight, MOI “decoupled” • shell thickness, added weight • fatter barrel, thinner handle • Weight distribution more uniform • ICM larger (less rot. recoil) • Ihandle smaller (easier to swing) • less mass at contact point • Dynamics • Stiffer for bending • Less energy lost due to vibrations • More compressible • COReff larger**tennis ball/racket**Effect of Bat on COR: Local Compression • CM energyshared between ball and bat • Ball inefficient: 75% dissipated • Wood Bat • kball/kbat ~ 0.02 • 80% restored • eeff = 0.50-0.51 • Aluminum Bat • kball/kbat ~ 0.10 • 80% restored • eeff = 0.55-0.58 Ebat/Eball kball/kbat xbat/ xball >10% larger!**Wood versus Aluminum:**Dynamics of “Trampoline” Effect “bell” modes: “ping” of bat • Want k small to maximize stored energy • Want >>1 to minimize retained energy • Conclusion: there is an optimum **Things I would like to understand better**• Relationship between bat speed and bat weight and weight distribution • Location of “physiological” sweet spot • Better model for the ball • Better understanding of trampoline effect for aluminum bat • Why is softball bat different from baseball bat? • Effect of “corking” the bat**Summary & Conclusions**• The essential physics of ball-bat collision understood • bat can be well characterized • ball is less well understood • the “hands don’t matter” approximation is good • Vibrations play important role • Size, shape of bat far from impact point does not matter • Sweet spot has many definitions • Aluminum outperforms wood!